Restructuring of a Model Hydrophobic Surface: Monte Carlo

Aug 20, 2013 - ABSTRACT: A lattice model is proposed to explain the restructuring of an ionic surfactant absorbed on a charged surface. When immersed ...
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Restructuring of a Model Hydrophobic Surface: Monte Carlo Simulations Using a Simple Coarse-Grained Model Changsun Eun,† Jhuma Das,‡ and Max L. Berkowitz* Department of Chemistry, University of North Carolina at Chapel Hill, Chapel Hill, North Carolina 27599, United States ABSTRACT: A lattice model is proposed to explain the restructuring of an ionic surfactant absorbed on a charged surface. When immersed in water, an ionic mica plate initially covered by a monolayer of surfactants rearranges to a surface inhomogeneously covered by patches of surfactant bilayer and bare mica. The model considers four species that can cover lattice sites of a surface. These species include (i) a surfactant molecule with its headgroup down, (ii) surfactant molecule with the headgroup up, (iii) a surfactant dimer arranged in a tail-to-tail configuration, which is a part of a bilayer, and (iv) a mica lattice site covered by water. We consider that only nearest neighbors on the lattice interact and describe the interactions by an interaction matrix. Using this model, we perform Monte Carlo simulations and study how the structure of the inhomogeneous surface depends on the interaction between water covered lattice site and its neighboring surfactant species covered sites. We observe that when this interaction is absent, the system undergoes phase separation into a bilayer phase and mica surface covered with water. When this interaction is taken into account, patches of surfactant bilayer and water are present in our system. The interaction between mica surfaces covered by patches of ionic surfactants is studied in experiments to understand the nature of long-ranged “hydrophobic” forces.

I. INTRODUCTION Water in nature is mostly found in a confined environment. Its properties are strongly influenced by the presence of surfaces that create the confinement. To study the structure and dynamics of the interfacial water, sophisticated spectroscopic techniques are used.1 In its turn the restructured water is responsible for the existence of interfacial interactions that are specifically due to water. Among such interactions, perhaps the best-known one is the hydrophobic interaction, responsible for the attraction between surfaces.2 It is expected that the range of the hydrophobic interactions will be few nanometers, since this is a distance range over which water structural and dynamical properties are influenced by the presence of the surface.1 Nevertheless, in the early 1980s, a long-range (above 20 nm) attraction between mica surfaces covered with CTAB (cetyl trimethyl ammonium bromide) surfactant monolayers was discovered.3 Since these surfaces were considered to represent nice examples of hydrophobic surfaces, the interaction was called long-ranged “hydrophobic” interaction, although the nature of its range remained puzzling. Following the initial discovery of the long-range attraction between surfaces immersed in water, this interaction has been extensively studied using both experimental and theoretical tools applied to a variety of different systems.4−20 As a result of these efforts, few mechanisms have been proposed to explain the origin of this interaction: (i) bridging submicroscopic bubbles between hydrophobic surfaces, (ii) cavitation in the intervening fluid, and (iii) electrostatic interaction between charged domains of the restructured hydrophobic surfaces. On the basis of the recent work that used atomic force microscopy and surface balance apparatus, the third mechanism seems to be the one © 2013 American Chemical Society

that explains the long-range interaction between mica surfaces coated with ionic surfactants and immersed in water.10,13,17 A hydrophobic surface can be created when a mica plate exposed to air is covered with a homogeneous monolayer of ionic surfactants so that the headgroups of the surfactants undergo physisorption to mica surface. When the surfactantcovered plate is immersed in water, the monolayer undergoes rearrangement; some of the surfactant molecules reorient, and eventually a surfactant bilayer is created, while water covers empty spots created on mica surface. Therefore, the mica surface previously homogeneously covered by surfactants turns into an inhomogeneous surface, containing positively charged bilayer patches due to surfactants and negatively charged bare mica surface patches.8 Upon approaching each other, the two inhomogeneous surfaces containing charged patches interact resulting in a long-ranged attraction.10,13,17 The mechanism of this long-range attraction is still not clear: it may be due to the free energy of counterions located in solution between the surfaces, or it may be due to the correlated arrangement of the patches.17 Since the restructuring of the surface plays a crucial role in understanding the nature of the long-ranged interaction between surfactant-covered mica surfaces, we would like to understand how such restructuring occurs. All atom computer simulations often provide molecular details of occurring phenomena that are hard, if not impossible, Special Issue: Michael D. Fayer Festschrift Received: June 17, 2013 Revised: August 20, 2013 Published: August 20, 2013 15584

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Figure 1. Schematic diagram of four possible states in our lattice model. The magenta rectangle represents negatively charged mica surface, and the surfactants are shown with a green sphere (headgroup) and a blue tail.

II. MODEL AND SIMULATION METHODS (i) Model System. In our model the surfactant-coated negatively charged mica surface was represented as a twodimensional (2D) lattice. Each lattice site was initially assigned to a predefined state, designated as state d or headgroup-down state representing a lattice site occupied by a single surfactant with its headgroup atoms facing mica surface and its tail directed away. According to the experimental observations and our MD simulation results, the d-state surfactants can undergo one of the two following transitions, depending on their orientations/conformations: (i) surfactants can flip, and as a result, their tails face the surface (surfactants move to a state with the headgroup-up or an u state); (ii) two neighboring surfactant monomers in the headgroup-down and headgroupup states can reorganize to form a dimer corresponding to a unit of a bilayer (state b). As dictated by the surfactant number conservation, because of the dimer formation on one site, an empty cell is formed at the site from which the surfactant is removed. The site devoid of a surfactant corresponds to the negatively charged mica site exposed to water molecules, and correspondingly the state is called water filled state (w). Thus, on the basis of the described conformational changes, our model comprises lattice sites that can be occupied by four states: (i) headgroup-down (d), (ii) headgroup-up (u), (iii) dimer or bilayer (b), and (iv) water-filled (w). Figure 1 shows the schematic representation of the four states. Our goal is to construct the simplest possible coarse-grained model; therefore, we considered only the interactions between states that are nearest neighbors on the lattice. Thus, we assumed that the interactions between sites are effectively short ranged due to cancellation of charges making up a site and also screening of charges by counterions that are located in the vicinity of mica and surfactants. Because of the existing four possibilities, the lattice of N sites can be in 4N different configurations. The energy of each configuration can be calculated using the following Hamiltonian,

to get from experiment. Thus, to study the restructuring of a small patch of surfactants on the mica surface in molecular details, we recently performed molecular dynamics simulations using a united atom force field.14 In our study the initial unit cell comprised a layer of mica, a layer of surfactant molecules (RN(CH3)3+Cl−, where R is a hydrocarbon chain) physisorbed by their headgroups to the mica surface, and water placed above the hydrophobic surface created by the surfactant tails. We observed that a homogeneous monolayer of surfactants transformed initially into a cylindrical micelle and finally into a spherical micelle. Because simulations with a detailed molecular description require substantial computer resources, it is not possible today to reproduce by detailed simulations the sizes of patches that are observed in experiments on mica surfaces covered with ionic surfactant. It is also not possible to run detailed simulations over periods of time required to explore the surface restructuring phenomenon. To study the surface restructuring on time and length scales that are consistent with the experimental data, one needs to use coarse-grained approach. One of the popular approaches to coarse-graining of amphiphilic molecules is an approach where a group of atoms is replaced by a particle that preserves prominent features of the replaced group. This approach is used to create force fields such as MARTINI, which is successfully used to simulate properties of phospholipid bilayers.21 Although simulations with a MARTINI-type force field can reach today a relatively long time scale, still it will be short compared to the time scale explored in experiments on surface restructuring. It seems that there is a strong need to develop simple coarse-grained models that will illuminate the nature of surface restructuring phenomenon. Therefore, it is the objective of this paper to demonstrate that such a model can be created. We accomplished the task by using a 2D lattice surface covered by just four different “species”. Monte Carlo simulations were performed on our reduced model to obtain structures that were energetically stable. Using the lattice model approach and thus compromising the detailed structure, we nevertheless were able to reproduce the gross features of the restructuring of the ionic surfactant monolayer that initially covered mica surface.

⎛ H = kBT ⎜⎜ ∑ Kα + ⎝ α 15585

⎞

∑ Jα ,β ⎟⎟ α,β

⎠

(1)

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Table 1. Coupling Constant Matrix J for All of the Pairs of Speciesa state

headgroup-down (d)

headgroup-up (u)

dimer (b)

water (w)

headgroup-down (d) headgroup-up (u) dimer (b) water (w)

Jdd = 1.0 Jud = 0.5 Jbd = 1.0 Jwd = 0

Jdu = 0.5 Juu = 1.0 Jbu = 0.5 Jwu = 0

Jdb = 1.0 Jub = 0.5 Jbb = 2.0 Jwb = 0

Jdw = 0 Juw = 0 Jbw = 0 Jww = 1.0

a

The interaction strengths Jαβ of all interacting pairs are relative and are obtained by taking the ratio of the average interaction energy of an interacting pair to the average d−d species interaction energy (Edd ≈ −17kBT). The value of +2 means strong interaction and +1 intermediate attraction, and 0.5 is used to model weak attraction.

Table 2. Coupling Constant Matrix J for All of the Pairs of Species, with the Values of Interaction between w and Other Species Not Equal to Zeroa state

headgroup-down (d)

headgroup-up (u)

dimer (b)

water (w)

headgroup-down (d) headgroup-up (u) dimer (b) water (w)

Jdd = 1.0 Jud = 0.5 Jbd = 1.0 Jwd = 0.5

Jdu = 0.5 Juu = 1.0 Jbu = 0.5 Jwu = −0.5

Jdb = 1.0 Jub = 0.5 Jbb = 2.0 Jwb = 0.5

Jdw = 0.5 Juw = −0.5 Jbw = 0.5 Jww = 1.0

a The interaction strengths Jαβ of all interacting pairs are relative and are obtained by taking the ratio of the average interaction energy of an interacting pair to the average d−d species interaction energy (Edd ≈ −17kBT). The value of +2 means strong interaction and +1 intermediate attraction, 0.5 weak attraction and −0.5 is used to model weak repulsion.

where Kα represents the self-energy of the state α due to cost/ gain in energy of the site occupied by one of the four states independent of this site’s neighbors. Jαβ is a coupling energy between a site in the α state and a site in the β state, and the summation is applied to nearest neighbor site pairs only. Since α and β in Jαβ can be in one of the four states, we need to consider a 4-by-4 matrix for J. Because of the symmetry of the J-matrix, only 10 parameters are required to describe it. With the additional four parameters for self-energy values, the Hamiltonian for the system given by eq 1 requires specification of 14 parameters. To reduce the number of these parameters, we neglected the terms with self-energy. We argue that the values of self-energy interactions are small compared to the values of nearest-neighbor interactions. First of all, the values of self-energy and nearest neighbor interaction energies should scale proportionally to the surface over which these interactions occur. The surface over which nearest-neighbors interact is larger than the surface exposed for self-interaction, which corresponds to headgroup and tail vertical area. Moreover, for site d the self-energy is due to surfactant headgroup interacting with mica and surfactant tail interacting with water; while the former is favorable, the latter is unfavorable. If we assume that the two interactions are of similar magnitude and effectively cancel each other out, the resulting self-interaction is small. The same argument can be made for the u state, and little change in self-energy is expected for a transition from d state to u. A change in self-energy will happen in transitions from d or u states to b or w, but because of scaling with surface area argument from above, we assume that it is small compared to change in the site−site interaction energy. Once the self-energy terms are neglected, the Hamiltonian of the system simplifies to the following form: H = kBT ∑ Jα , β α ,β

Thus, the interaction between dd, uu, and ww states were assumed to have the same value, while the interaction between the bb states, which involves an interaction between two surfactant chains, was assumed to be twice as strong as the reference dd interaction. The interaction of the u state with its neighbors d and b was penalized and assigned half of the value of the reference dd interaction, while the interaction between d and b states was considered to be the same as between two d states, since only interaction between one chain from b and one chain from d was involved in this case. From our molecular dynamics simulations we observed that water surfactant interactions (monomer or dimer) are not that strong, and therefore, we initially neglected these interactions (see Table 1). To understand how the results depend on the values of these water−surfactant interactions, we considered another case where the interactions were not neglected and their numerical values were also considered to be equal to half of the values of the reference dd interactions (see Table 2). As we shall demonstrate below, that produced a substantial difference in the stable structures. Thus, on the basis of a short detailed molecular dynamics simulation and using some plausible physical arguments, we reduced the problem of assigning 10 parameters for matrix J to just finding one parameter, a reference Jdd interaction. The value of this reference interaction, mentioned in Table 1, was obtained from a short detailed simulation. (ii) Monte Carlo Simulations. We performed Monte Carlo (MC) simulations using the Hamiltonian given by eq 2 to see if our simple model can reproduce qualitative features of the experimental results. As in the surface-restructuring experiments, we prepared an initial configuration as a monolayercoated surface. Thus, initially our model surface (Figure 2a) was represented by a 2D lattice consisting of N2 (N = 86) lattice sites with each site occupied by a surfactant monomer in a d state. To connect with the dimensions of our problem, we assume that the area per lattice site is equal to 0.58 × 0.58 nm2, the value determined from the area of the surfactant headgroup in an atomistic model. This gives us the size of the model surface that is equal to ∼50 × 50 nm2.

(2)

To get the relative values for the elements of matrix J, we were guided by arguments dictated by the physics of the system. Since we wanted to build a simplest coarse-grained model, we decided to use approximate values of Jαβ that reflected the relative strength of interactions between the states. 15586

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state with a stable value of energy, we calculated the total energy of the system as a function of the number of MC steps. We considered the system to be in a stable state when the total energy of the system remained stable.

III. RESULTS To find out if and when our systems reached stability during MC simulations, we calculated the total energy (in units of kBT) as a function of number of MC steps. In the upper panel of Figure 3 we show how the total energy changes with the Figure 2. Initial configuration of 2D lattice model for study of the surface restructuring phenomena (left) and the system conformation after 100 MC steps of MC simulation for system I. This configuration was used as the initial state for the equilibration run. As required by the experimental setup, the starting configuration (left) is a homogeneous hydrophobic surface, where the headgroup atoms of the surfactants are facing the lattice surface and the corresponding tails are directed away from the surface (state d). The system size is ∼50 × 50 nm2 = 2500 nm2. Cyan, yellow, red and blue colors represent states d, u, b, w, respectively.

In our MC simulations we performed the following MC moves or transitions. (a) For flipping of a monomer, we considered that when a particular site was in a d state, it was allowed to transit into an u state. If it was in an u state, it was allowed to change into a d state. (b) For monomer−dimer transition, a particular site and one of its neighboring sites were considered, and when the two neighboring states were d and u, these two lattice sites could transform so that the site with the state d could turn into state b (dimer) and the other site could become a w site (site filled with water). Similarly, when the two neighboring states were b and w, the dimer could break down and regenerate d and u states at the two neighboring sites. (c) For efficient sampling, additionally a global exchange transition was implemented in the model so that any two sites on the lattice were selected randomly and were allowed to exchange place with one another. The transitions described above constitute one MC elementary move in our simulation. The trial configurations from all the transitions described above were accepted or rejected using the standard Metropolis algorithm.22 The configuration with a pure monolayer covering the lattice represents a very high-energy state, so we first carried out a short MC simulation to prepare a more meaningful initial configuration. Thus, we started with all lattice sites in d states and subjected the system to 100 MC steps. Each MC step consisted of 1000 elementary MC moves described above performed on randomly chosen sites. After this initial process, the system reached a lower energy configuration, which was considered to be an initial one (see Figure 2b). Staring from this initial configuration, we performed two sets of MC production runs: one with the parameters for matrix J taken from Table 1 (system I) and the other with the parameters for matrix J given in Table 2 (system II). Each set contained three runs, but we present the results from one of the runs only, since the results are qualitatively very similar. We ran the simulations for the system I for 200 000 steps, while the runs for system II were done for 400 000 steps. For each of the MC step, N2 elementary MC moves were performed by moving over all lattice sites. Since we used a lattice model, during the entire simulation, the total number (N) of sites and total area (A) of the system were kept constant. To examine whether the system reached a

Figure 3. Upper panel: total energy as a function of Monte Carlo steps during the equilibrium simulation run for system I. Total energy levels off and attains a steady value after 110 000 MC steps. Lower panel: total energy as a function of Monte Carlo steps during the equilibrium simulation run I for system II. Total energy levels off and attains a steady value after 200 000 MC steps.

number of MC steps in the case of simulations performed on system I with the coupling parameters taken from Table 1. From this figure we can see that energy decreased rapidly for the first 20 000 MC steps, meaning that the system had undergone major structural rearrangements. After that, the energy decreased slowly and reached a constant plateau at around 110 000 MC steps; this leveling off indicates that the system reached a stable state. Similarly, the total energy of the system with parameters from Table 2 (system II) is presented in the lower panel of Figure 3. Again the initial decrease of energy was rapid, but after 20 000 MC steps it slowed, and after 200 000 MC steps the energy reached the saturation value. Note that the saturation energy for system II was higher than for system I. We observed that in our MC simulations, systems I and II efficiently evolved toward stable states. The systems started their development as surfaces initially covered by a homogeneous mixture of four different species (Figures 4a and 5a) and evolved into surfaces covered with a heterogeneous mixture of mostly b and w species. The structural evolution of systems I and II during our MC simulations is shown in Figures 4 and 5. In the beginning of the simulations, more and more of d and u neighbors converted into a neighboring pair of b and w, since the coupling energy between two neighboring dimers is twice as attractive as the interaction between neighboring d or u monomers. For the system I we observed that during the first 10 000 MC steps a large number of small but interconnected 15587

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Figure 4. Structural evolution of the lattice model system during equilibration of system I. The homogeneous lattice surface transforms into a heterogeneous charged surface with a patch made of the surfactant dimer states b (red region) and a patch created by w species (blue region).

islands of b (bilayer patches) and w (water domains) species were generated spontaneously and coexisted with tiny clusters of d and u species in a highly miscible state (Figure 4b). As the simulations progressed, nanometer sized surfactant bilayer patches and water filled domains were created until the aggregates reached stable sizes (Figure 4c−f). The stable state the system I reached after around 100 000 MC steps was a state with approximately half the lattice area covered with a large bilayer patch (Figure 4g, red region), and the other half was filled with water (Figure 4g, blue region), although some impurities in the form of tiny clusters of d and u species in these two domains could be easily observed. Considering that the systems in our simulations have periodic boundary conditions, it is easy to realize that a stable state of system is a state when a complete phase separation of the surfactant bilayer and water has occurred. During the last 100 000 MC steps very small change in the structure was observed, although the boundary region between phases got slightly reorganized. For system II, the structural transition of the lattice surface is depicted in Figure 5. For this system the interaction energies among the intra- and interspecies pairs were mostly the same as for system I (see Tables 1 and 2) except that the interaction energy strengths for d:w, u:w and b:w pairs were set to non-zero values, while these were equal to 0 for system I. Unlike in the run with parameters for system I, where we observed a complete phase separation between the surfactant bilayer and water, we observed that to reach a stable state of system II

required a larger number of MC steps (∼200 000 MC steps) and that in a stable state a number of disconnected domains containing b and w species were still present. The existence of such domains is probably due to the presence of an attractive interaction between w and b species (Jwb in system II describes attraction); the attractive Jwb reduces the hydrophobic character of the bilayer species and thus reduces the line tension between the b and w domains.

IV. SUMMARY AND CONCLUSIONS In this paper we presented a lattice model that describes a restructuring process when a mica surface initially homogeneously covered by a surfactant monolayer transforms into an inhomogeneous surface containing patches of surfactant bilayers and patches of bare mica. By applying our model, we could simulate a nanoscopic sized system (50 × 50 nm2) and observe structural transitions that take place for a surfactant monolayer covering mica surface. We used MC simulations on a system subject to Hamiltonian given by eq 2 and considered two sets of parameters. The difference between the sets was in the treatment of water−surfactant interactions. In system I we neglected this interaction (we considered that attraction between water and headgroup is canceled by the “hydrophobic” repulsion between water and the surfactant tails). In system II the interaction between water and surfactant species was taken into account. In our MC simulation runs of system I the model surface structurally transformed into mostly two patches: a 15588

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lipids (one of them is cholesterol) are mixed in a lipid bilayer.26,27 In experiments like the ones performed in Klein’s group, the patches carry charges (surfactant bilayer patches are positively charged, while bare mica are negatively charged), although the patches are strongly shielded by counterions. For a long time the long-ranged “hydrophobic” attraction between charged patched surfaces was considered to be due to correlated arrangement between patches, positive patches face negative patches and vice versa. Very recently it was proposed that such arrangement is not needed to observe long ranged hydrophobic attraction.17 To study in more detail the nature of interaction between charged patched surfaces, one would like to perform computer simulations using coarse-grained models of the type recently used in simulations by Jho et.al.15 Our simplistic model can be used in future coarse-grained studies, for example, to understand how the strength of the interaction between surfactants or temperature determines the sizes of the patches or the phase behavior of the system. This issue can be understood when a large number of different simulations will be done to map out the phase diagrams of systems described by Hamiltonians from eqs 1 and 2.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Present Addresses †

Howard Hughes Medical Institute, University of California at San Diego, La Jolla, California, 92093, USA. ‡ Department of Biochemistry and Biophysics, University of North Carolina at Chapel Hill, Chapel Hill, North Carolina, 27599, USA. Notes

Figure 5. Structural evolution of the lattice model system during equilibration of system II. The homogeneous lattice surface transforms into a heterogeneous charged surface with a patch made of the surfactant dimer states b (red region) and a patch created by w species (blue region).

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS C.E. and J.D. contributed equally to the manuscript. The support by Grant N000141010096 from the Office of Naval Research is gratefully acknowledged.

■

patch of a bilayer and a patch of mica, indicating a full phase separation in this case. In simulation with system II we also observed that one large-sized bilayer and one large-sized water domain were present in the simulation. In addition, smaller sized domains were also present. The domain size of the large bilayer patch was approximately 30 nm, smaller than the domain size reported in the experiments (∼150 nm). Simulations of model surfaces of bigger dimensions probably could achieve the domain sizes of values similar to the ones seen in the experiments, but we observed that our present version of the program slowed substantially with an increase of memory demands when the size of the system increased beyond N ≈ 90. An exact quantitative agreement with experiment was not the goal of our present work; we only wanted to demonstrate that a simple four-state model is able to illustrate creation of patches of surfactant bilayers from an initial homogeneously absorbed surfactant monolayer. Our model could also complement or extend existing lattice models23−25 for the study of another interesting problem, the problem of a lipid raft. The main issue of the lipid raft problem is to understand the mechanism of domain creation and domain sizes when at least three different

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